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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
$\text{if y} =\text{x}^{x},\text{prove that } \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} -\frac{1}{\text{y}}\bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{2} -\frac{\text{y}}{\text{x}} =0.$
✓

Answer

$\text{y} = \text{x}^{\text{x}}\therefore\log\text{ y } =\text{x}\log\text{x},$
Taking log of both sides
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}} =\log\text{x} + 1 ,$
Diff. w r t “x”
$\Rightarrow\frac{1}{\text{y}}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} -\frac{1}{\text{y}^{2}}\bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{2} = \frac{1}{\text{x}},$
Diff. w r t “x”
$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} -\frac{1}{\text{y}}\bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{2} - \frac{\text{y}}{\text{x}} - 0 .$

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